Matched-pair t-test

Description

The t-test gives an indication of how separate two sets of measurements are,
allowing you to determine whether something has changed and there are two
distributions, or whether there is effectively only one distribution.

The matched-pair t-test (or paired t-test or paired samples
t-test or dependent t-test) is used when the data from the two groups
can be presented in pairs, for example where the same people are being measured
in before-and-after comparison or
when the group is given two different tests at different times (eg. pleasantness
of two different types of chocolate).

Goodness of fit

This can also be used when you have one measure and are matching against a
particular frequency distribution, for which you can determine 'should'
measures. The two most common distributions to test for are normal (bell-shaped)
and flat.

In a flat distribution, all items are equally likely. The t-test can be used
here to discover whether any one or more of a set of measures is significantly
different from the others.

This use of the chi-square test is often known as the 'Goodness of Fit' test.

Calculation

The value of t may be calculated using packages such as SPSS. The actual
calculation is:

t = AVERAGE(X1-X2) / (
Sd / SQRT( n) )

Where Sd is the standard deviation of
the differences and n is the number of pairs.

Sd = SQRT( (SUM((X1-X2)2)
- (SUM(X1-X2))2/n) / (n-1) )

Interpretation

The resultant t-value is then looked up in a t-table as below
to determine the probability that a significant difference between the two sets
of measures exists and hence what can be claimed about the efficacy of the
experimental treatment.

The t-value can also be interpreted as an r-value, which can be calculated as: